Spin-projection-protected crossings in an exchange spectrum

In the two-spin exchange spectrum with unequal signed magnetic moments, the axial Hamiltonian conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm \hslash$, while the two mixed states lie in the $M=0$ block. Levels in different $M$ sectors may cross as the axial magnetic field changes because the conserved projection forces their coupling matrix element to vanish.

This is a symmetry qualification of the non-crossing rule. A nonzero isotropic exchange interaction does not by itself turn every field-dependent crossing into an avoided crossing. A transverse perturbation that breaks conservation of $M$ can mix the sectors and open a gap, provided its relevant matrix element is nonzero. Plotting the energy eigenvalues by their conserved projection therefore distinguishes protected crossings from the avoided splitting already present within the $M=0$ block.